Abstract Fractional Monotone Approximation with Applications

Abstract

Here we extended our earlier fractional monotone approximation theory to abstract fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let $f\in C^p\left(\left[-1,1\right]\right)$, $p\ge 0$ and let L be a linear abstract left or right fractional differential operator such that $L\left(f\right)\ge 0$ over $\left[0,1\right]$ or $\left[-1,0\right]$, respectively. We can find a sequence of polynomials $Q_n$ of degree $\le n$ such that $L\left(Q_n\right)\ge 0$ over $\left[0,1\right]$ or $\left[-1,0\right]$, respectively. Additionally f is approximated quantitatively with rates uniformly by $Q_n$ with the use of first modulus of continuity of $f^{\left(p\right)}$.

1. Introduction

The topic of monotone approximaton initiated in 1965, Ref. [1] by O. Shisha, and became a major trend in approximation theory. The original problem was: given a positive integer k, approximate with rates a given function f whose kth derivative is ≥0 by polynomials ${\left(Q_n\right)}_{n\in \mathbb{N}}$ having the same property.

In 1985, Ref. [2], the author and O. Shisha continued this study by replacing the kth derivative with a linear differential operator of order k involving ordinary derivatives, again the approximation was with rates.

Later, in 1991, Ref. [3], the author extended this kind of study in two dimensions, etc.

In 2015, Ref. [4] (see chapters 1–8) went a step further, by starting the fractional monotone approximation, in that the linear differential operator is a fractional one, involving left or right side Caputo fractional derivatives.

To give a flavor of it, we need:

Definition 1

([5], p. 50). Let $\alpha >0$ and $⌈\alpha ⌉=m\in \mathbb{N}$ ($⌈·⌉$ is the ceiling of the number). Consider $f\in C^m\left(\left[-1,1\right]\right)$. We define the left side Caputo fractional derivative of f of order α as follows:

$$\left(D_{*-1}^{\alpha }f\right)\left(x\right)=\frac{1}{\mathsf{\Gamma }\left(m-\alpha \right)}{\int }_{-1}^x{\left(x-t\right)}^{m-\alpha -1}{f}^{\left(m\right)}\left(t\right)dt,$$

(1)

for any $x\in \left[-1,1\right]$, where Γ is the gamma function. We set

$$D_{*-1}^{0}f\left(x\right)=f\left(x\right),D_{*-1}^{m}f\left(x\right)=f^{\left(m\right)}\left(x\right),\forall x\in \left[-1,1\right].$$

In addition, to motivate our work, we mention:

Theorem 1

([4], p. 2). Let $h,k,p$ be integers, $0\le h\le k\le p$ and let f be a real function, with $f^{\left(p\right)}$ continuous in $\left[-1,1\right]$ and first modulus of continuity ${\omega }_1\left(f^{\left(p\right)},\delta \right)$, where $\delta >0$. Let ${\alpha }_j\left(x\right)$, $j=h,h+1,\dots ,k$ be real functions, defined and bounded on $\left[-1,1\right]$ and assume for $x\in \left[0,1\right]$ that ${\alpha }_h\left(x\right)$ is either ≥ some number $\alpha >0$ or ≤ some number $\beta <0$. Let the real numbers ${\alpha }_0=0<{\alpha }_1\le 1<{\alpha }_2\le 2<\dots <{\alpha }_p\le p$. Here $D_{*-1}^{{\alpha }_j}f$ stands for the left Caputo fractional derivative of f of order ${\alpha }_j$ anchored at $-1$. Consider the linear left fractional differential operator

$$L:=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[D_{*-1}^{{\alpha }_j}\right]$$

(2)

and suppose, throughout $\left[0,1\right],$

$$L\left(f\right)\ge 0.$$

(3)

Then, for any $n\in \mathbb{N}$, there exists a real polynomial $Q_n\left(x\right)$ of degree $\le n$ such that

$$L\left(Q_n\right)\ge 0\mathit{throughout}\left[0,1\right],$$

(4)

and

$$\underset{-1\le x\le 1}{max}\left|f\left(x\right)-Q_n\left(x\right)\right|\le C{n}^{k-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(5)

where C is a constant independent of n or $f.$

As you see the monotonicity property here is true only on the critical interval $\left[0,1\right].$

We will use the following important result:

Theorem 2

(see: [6] by S.A. Teljakovskii and [7] by R.M. Trigub). Let $n\in \mathbb{N}$. Be given a real function g, with $g^{\left(p\right)}$ continuous in $\left[-1,1\right]$, there exists a real polynomial $q_n\left(x\right)$ of degree $\le n$ such that

$$\underset{-1\le x\le 1}{max}\left|g^{\left(j\right)}\left(x\right)-q_n^{\left(j\right)}\left(x\right)\right|\le R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right),$$

(6)

$j=0,1,\dots ,p$, where $R_p$ is a constant independent of n or g.

In this article, we perform abstract fractional calculus, left and right monotone approximation theory of Caputo type, and then we apply our results to Prabhakar fractional Calculus, generalized non-singular fractional calculus, and parametrized Caputo-Fabrizio non-singular fractional calculus.

Next, we build the related necessary fractional calculi background.

2. Fractional Calculi

Here, we need to be very specific in preparation for our main results.

2.1. Abstract Fractional Calculus

Let $h,k\in {\mathbb{Z}}_{+}$, $p\in \mathbb{N}:0\le h\le k\le p$. Let also $\mathbb{N}/\ni {\alpha }_j>0$, $j=1,\dots ,p$, such that ${\alpha }_0=0<{\alpha }_1<1<{\alpha }_2<2<{\alpha }_3<3<\dots <\dots <{\alpha }_p<p$. That is $⌈{\alpha }_j⌉=j$, $j=1,\dots ,p$; $⌈{\alpha }_0⌉=0$.

Consider the integrable functions $k_j:=K_{{\alpha }_j}:\left[0,2\right]\to {\mathbb{R}}_{+}$, $j=0,1,\dots ,p$. Here, $g\in C^p\left(\left[-1,1\right]\right).$

We consider the following abstract left side Caputo type fractional derivatives:

$$\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}g\right)\left(x\right):={\int }_{-1}^x{k}_j\left(x-t\right)g^{\left(j\right)}\left(t\right)dt,$$

(7)

$j=1,\dots ,p;$ ∀$x\in \left[-1,1\right].$

Similarly, we define the corresponding right side generalized Caputo type fractional derivatives:

$$\left({}^{k_j}{D}_{1-}^{{\alpha }_j}g\right)\left(x\right):={\left(-1\right)}^j{\int }_x^1{k}_j\left(t-x\right)g^{\left(j\right)}\left(t\right)dt,$$

(8)

$j=1,\dots ,p;$ ∀$x\in \left[-1,1\right].$

We set

$$\left({}^{k_j}{D}_{*-1}^{j}g\right)\left(x\right):=g^{\left(j\right)}\left(x\right);\left({}^{k_j}{D}_{1-}^{j}g\right)\left(x\right):={\left(-1\right)}^j{g}^{\left(j\right)}\left(x\right),$$

(9)

for $j=1,\dots ,p$, and also we set

$$\left({}^{k_0}{D}_{*-1}^{0}g\right)\left(x\right):=\left({}^{k_0}{D}_{1-}^{0}g\right)\left(x\right):=g\left(x\right),$$

(10)

∀$x\in \left[-1,1\right].$

We will assume that

$${\int }_0^1{k}_h\left(z\right)dz\ge 1,\mathrm{when}h\ne 0.$$

(11)

In the usual Caputo fractional derivatives case, it is

$$k_j\left(z\right)=\frac{z^{j-{\alpha }_j-1}}{\mathsf{\Gamma }\left(j-{\alpha }_j\right)},j=1,\dots ,p;\forall z\in \left[0,2\right],$$

(12)

and (11) is fulfilled, by the fact that $\mathsf{\Gamma }\left(h-{\alpha }_h+1\right)\le 1$, see [4], p. 6.

2.2. About Prabhakar Fractional Calculus

Here, we follow [8,9].

We consider the Prabhakar function (also known as the three parameter Mittag-Leffler function), (see [10], p. 97; [11])

$$E_{\alpha ,\beta }^{\gamma }\left(z\right)=\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k}{k!\mathsf{\Gamma }\left(\alpha k+\beta \right)}{z}^k,$$

(13)

where $\mathsf{\Gamma }$ is the gamma function; $\alpha ,\beta >0,$ $\gamma \in \mathbb{R},$ $z\in \mathbb{R},$ and ${\left(\gamma \right)}_k=\gamma \left(\gamma +1\right)...\left(\gamma +k-1\right)$. It is $E_{\alpha ,\beta }^0\left(z\right)=\frac{1}{\mathsf{\Gamma }\left(\beta \right)}.$

Let $a,b\in \mathbb{R}$, $a<b$ and $x\in \left[a,b\right]$; $f\in C\left(\left[a,b\right]\right).$ The left and right Prabhakar fractional integrals are defined ([8,9]) as follows:

$$\left(e_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(x\right)={\int }_a^x{\left(x-t\right)}^{\mu -1}{E}_{\rho ,\mu }^{\gamma }\left[\omega {\left(x-t\right)}^{\rho }\right]f\left(t\right)dt,$$

(14)

and

$$\left(e_{\rho ,\mu ,\omega ,b-}^{\gamma }f\right)\left(x\right)={\int }_x^b{\left(t-x\right)}^{\mu -1}{E}_{\rho ,\mu }^{\gamma }\left[\omega {\left(t-x\right)}^{\rho }\right]f\left(t\right)dt,$$

(15)

where $\rho ,\mu >0;$ $\gamma ,\omega \in \mathbb{R}.$

Functions (14) and (15) are continuous, see [8].

Next, let $f\in C^N\left(\left[a,b\right]\right)$, where $N=⌈\mu ⌉$, ($⌈·⌉$ is the ceiling of the number), $0<\mu \notin \mathbb{N}$. We define the Prabhakar-Caputo left and right fractional derivatives of order $\mu$ ([8,9]) as follows ($x\in \left[a,b\right]$):

$$\left({}^C{D}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(x\right)={\int }_a^x{\left(x-t\right)}^{N-\mu -1}{E}_{\rho ,N-\mu }^{-\gamma }\left[\omega {\left(x-t\right)}^{\rho }\right]f^{\left(N\right)}\left(t\right)dt,$$

(16)

with ${}^C{D}_{\rho ,0,\omega ,a+}^{\gamma }f:=f;$ ${}^C{D}_{\rho ,N,\omega ,a+}^{\gamma }f:=f^{\left(N\right)}$, $N\in \mathbb{N},$ and

$$\left({}^C{D}_{\rho ,\mu ,\omega ,b-}^{\gamma }f\right)\left(x\right)={\left(-1\right)}^N{\int }_x^b{\left(t-x\right)}^{N-\mu -1}{E}_{\rho ,N-\mu }^{-\gamma }\left[\omega {\left(t-x\right)}^{\rho }\right]f^{\left(N\right)}\left(t\right)dt,$$

(17)

with ${}^C{D}_{\rho ,0,\omega ,b-}^{\gamma }f:=f;$ ${}^C{D}_{\rho ,N,\omega ,b-}^{\gamma }f:={\left(-1\right)}^N{f}^{\left(N\right)}$, $N\in \mathbb{N}.$

One can rewrite (16) and (17) as

$$\left({}^C{D}_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(x\right)=\left(e_{\rho ,N-\mu ,\omega ,a+}^{-\gamma }{f}^{\left[N\right]}\right)\left(x\right),$$

(18)

and

$$\left({}^C{D}_{\rho ,\mu ,\omega ,b-}^{\gamma }f\right)\left(x\right)={\left(-1\right)}^N\left(e_{\rho ,N-\mu ,\omega ,b-}^{-\gamma }{f}^{\left[N\right]}\right)\left(x\right),$$

(19)

∀$x\in \left[a,b\right]$.

Clearly, the functions (18) and (19) are continuous.

By [8], we have that

$$\left(e_{\rho ,\mu ,\omega ,a+}^{\gamma }f\right)\left(x\right)={\int }_a^x{\left(x-t\right)}^{\mu -1}\left(\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k}{k!\mathsf{\Gamma }\left(\rho k+\mu \right)}{\left(\omega {\left(x-t\right)}^{\rho }\right)}^k\right)f\left(t\right)dt$$

$$=\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+\mu \right)}{\int }_a^x{\left(x-t\right)}^{\left(\rho k+\mu \right)-1}f\left(t\right)dt,$$

(20)

∀$x\in \left[a,b\right]$.

That is

$$\left(e_{\rho ,\mu ,\omega ,a+}^{\gamma }1\right)\left(x\right)=\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+\mu \right)}{\int }_a^x{\left(x-t\right)}^{\left(\rho k+\mu \right)-1}dt=$$

$$\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+\mu \right)}\frac{{\left(x-a\right)}^{\left(\rho k+\mu \right)}}{\left(\rho k+\mu \right)}=\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+\mu +1\right)}{\left(x-a\right)}^{\left(\rho k+\mu \right)}=$$

(21)

$${\left(x-a\right)}^{\mu }\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k}{k!\mathsf{\Gamma }\left(\rho k+\mu +1\right)}{\left[\omega {\left(x-a\right)}^{\rho }\right]}^k={\left(x-a\right)}^{\mu }{E}_{\rho ,\mu +1}^{\gamma }\left(\omega {\left(x-a\right)}^{\rho }\right).$$

So, we have proved that

$$\left(e_{\rho ,\mu ,\omega ,a+}^{\gamma }1\right)\left(x\right)={\left(x-a\right)}^{\mu }{E}_{\rho ,\mu +1}^{\gamma }\left(\omega {\left(x-a\right)}^{\rho }\right),$$

(22)

∀$x\in \left[a,b\right]$.

Similarly, we have

$$\left(e_{\rho ,\mu ,\omega ,b-}^{\gamma }1\right)\left(x\right)=\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+\mu \right)}{\int }_x^b{\left(t-x\right)}^{\left(\rho k+\mu \right)-1}dt=$$

$$\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+\mu \right)}\frac{{\left(b-x\right)}^{\left(\rho k+\mu \right)}}{\left(\rho k+\mu \right)}=\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+\mu +1\right)}{\left(b-x\right)}^{\left(\rho k+\mu \right)}=$$

(23)

$${\left(b-x\right)}^{\mu }\sum _{k=0}^{\infty }\frac{{\left(\gamma \right)}_k}{k!\mathsf{\Gamma }\left(\rho k+\mu +1\right)}{\left[\omega {\left(b-x\right)}^{\rho }\right]}^k={\left(b-x\right)}^{\mu }{E}_{\rho ,\mu +1}^{\gamma }\left(\omega {\left(b-x\right)}^{\rho }\right).$$

So, it holds

$$\left(e_{\rho ,\mu ,\omega ,b-}^{\gamma }1\right)\left(x\right)={\left(b-x\right)}^{\mu }{E}_{\rho ,\mu +1}^{\gamma }\left(\omega {\left(b-x\right)}^{\rho }\right),$$

(24)

∀$x\in \left[a,b\right]$.

Next, we take $\left[a,b\right]=\left[-1,1\right].$

Thus,

$$\left(e_{\rho ,\mu ,\omega ,-1+}^{\gamma }1\right)\left(x\right)={\left(x+1\right)}^{\mu }{E}_{\rho ,\mu +1}^{\gamma }\left(\omega {\left(x+1\right)}^{\rho }\right),$$

(25)

and

$$\left(e_{\rho ,\mu ,\omega ,1-}^{\gamma }1\right)\left(x\right)={\left(1-x\right)}^{\mu }{E}_{\rho ,\mu +1}^{\gamma }\left(\omega {\left(1-x\right)}^{\rho }\right),$$

(26)

∀$x\in \left[-1,1\right]$.

Clearly, then we get

$$\left(e_{\rho ,N-\mu ,\omega ,-1+}^{-\gamma }1\right)\left(x\right)={\left(x+1\right)}^{N-\mu }{E}_{\rho ,N-\mu +1}^{-\gamma }\left(\omega {\left(x+1\right)}^{\rho }\right),$$

(27)

and

$$\left(e_{\rho ,N-\mu ,\omega ,1-}^{-\gamma }1\right)\left(x\right)={\left(1-x\right)}^{N-\mu }{E}_{\rho ,N-\mu +1}^{-\gamma }\left(\omega {\left(1-x\right)}^{\rho }\right),$$

(28)

∀$x\in \left[-1,1\right]$.

Here, it is $N-\mu >0$. By assumption we take $\rho >0$, $\gamma <0$ and, for convenience, we consider only $\omega >0$.

Therefore, we derive the basic Hardy type inequalities:

$${∥e_{\rho ,N-\mu ,\omega ,-1+}^{-\gamma }1∥}_{\infty ,\left[-1,1\right]}\le 2^{N-\mu }{E}_{\rho ,N-\mu +1}^{-\gamma }\left(2^{\rho }\omega \right),$$

(29)

and

$${∥e_{\rho ,N-\mu ,\omega ,1-}^{-\gamma }1∥}_{\infty ,\left[-1,1\right]}\le 2^{N-\mu }{E}_{\rho ,N-\mu +1}^{-\gamma }\left(2^{\rho }\omega \right).$$

(30)

2.3. From Generalized Non-Singular Fractional Calculus

We need

Definition 2.

Here, we use the multivariate analogue of generalized Mittag-Leffler function, see [12], defined for $\lambda ,{\gamma }_j,{\rho }_j,z_j\in \mathbb{C},$ $Re\left({\rho }_j\right)>0$ ($j=1,\dots ,m$) in terms of a multiple series of the form:

$$E_{\left({\rho }_j\right),\lambda }^{\left({\gamma }_j\right)}\left(z_1,\dots ,z_m\right)=E_{\left({\rho }_1,\dots ,{\rho }_m\right),\lambda }^{\left({\gamma }_1,\dots ,{\gamma }_m\right)}\left(z_1,\dots ,z_m\right)=$$

$$\sum _{k_1,\dots ,k_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{k_1}\dots {\left({\gamma }_m\right)}_{k_m}}{\mathsf{\Gamma }\left(\lambda +{\displaystyle \sum _{j=1}^m}{k}_j{\rho }_j\right)}\frac{z_1^{k_1}...z_m^{k_m}}{k_1!\dots k_m!},$$

(31)

where ${\left({\gamma }_j\right)}_{k_j}$ is the Pochhammer symbol, Γ is the gamma function. By [13], p. 157, (31) converges for $Re\left({\rho }_j\right)>0$, $j=1,\dots ,m$.

In what follows, we will use the particular case of $E_{\left(\rho ,...,\rho \right),\lambda }^{\left({\gamma }_1,\dots ,{\gamma }_m\right)}\left[{\omega }_1{t}^{\rho },\dots ,{\omega }_m{t}^{\rho }\right]$, denoted by $E_{\left(\rho \right),\lambda }^{\left({\gamma }_j\right)}\left[{\omega }_1{t}^{\rho },\dots ,{\omega }_m{t}^{\rho }\right]$, where $0<\rho <1$, $t\ge 0,$ $\lambda >0$, ${\gamma }_j\in \mathbb{R}$ with ${\left({\gamma }_j\right)}_{k_j}:={\gamma }_j\left({\gamma }_j+1\right)\dots \left({\gamma }_j+k_j-1\right)$, ${\omega }_j\in \mathbb{R}-\left\{0\right\}$, for $j=1,...,m$.

Let now $f\in C^{n+1}\left(\left[a,b\right]\right)$, $n\in {\mathbb{Z}}_{+}$.

We define the Caputo type generalized left fractional derivative with non-singular kernel of order $n+\rho$, as

$$D_{a*}^{n+\rho }f\left(x\right):={}_{\left({\gamma }_j\right)\left({\omega }_j\right)}^{CA}{D}_{a*}^{n+\rho ,\lambda }f\left(x\right):=$$

$$\frac{A\left(\rho \right)}{1-\rho }{\int }_a^x{E}_{\left(\rho \right),\lambda }^{\left({\gamma }_j\right)}\left[\frac{-{\omega }_1\rho }{1-\rho }{\left(x-t\right)}^{\rho },\dots ,\frac{-{\omega }_m\rho }{1-\rho }{\left(x-t\right)}^{\rho }\right]f^{\left(n+1\right)}\left(t\right)dt,$$

(32)

∀$x\in \left[a,b\right].$

Similarly, we define the Caputo type generalized right fractional derivative with non-singular kernel of order $n+\rho$, as

$$D_{b-}^{n+\rho }f\left(x\right):={}_{\left({\gamma }_j\right)\left({\omega }_j\right)}^{CA}{D}_{b-}^{n+\rho ,\lambda }f\left(x\right):=$$

$${\left(-1\right)}^{n+1}\frac{A\left(\rho \right)}{1-\rho }{\int }_x^b{E}_{\left(\rho \right),\lambda }^{\left({\gamma }_j\right)}\left[\frac{-{\omega }_1\rho }{1-\rho }{\left(t-x\right)}^{\rho },\dots ,\frac{-{\omega }_m\rho }{1-\rho }{\left(t-x\right)}^{\rho }\right]f^{\left(n+1\right)}\left(t\right)dt,$$

(33)

∀$x\in \left[a,b\right].$

Above $A\left(\rho \right)$ is a normalizing constant.

The above derivatives (32), (33) generalize the Atangana-Baleanu fractional derivatives [14].

We mention the following Hardy type inequalities:

Theorem 3

([15]). All as above with ${\gamma }_j>0$, $j=1,\dots ,m;$ $\lambda =1$. Then

$$\left\{{∥D_{a*}^{n+\rho }f∥}_{\infty },{∥D_{b-}^{n+\rho }f∥}_{\infty }\right\}\le \frac{\left(b-a\right)\left|A\left(\rho \right)\right|}{1-\rho }$$

$$E_{\left(\rho \right),2}^{\left({\gamma }_j\right)}\left[\frac{\left|{\omega }_1\right|\rho }{1-\rho }{\left(b-a\right)}^{\rho },\dots ,\frac{\left|{\omega }_m\right|\rho }{1-\rho }{\left(b-a\right)}^{\rho }\right]{∥f^{\left(n+1\right)}∥}_{\infty }<\infty ,$$

(34)

where $n\in {\mathbb{Z}}_{+}.$

We also mention:

Theorem 4

([15]). All as above with ${\gamma }_j>0$, $j=1,\dots ,m$, and $\lambda >0$, $0<\rho <1$, etc. Then

$$D_{a*}^{n+\rho }f,D_{b-}^{n+\rho }f\in C\left(\left[a,b\right]\right),n\in {\mathbb{Z}}_{+}.$$

We rewrite (32) and (33), and for $\left[a,b\right]=\left[-1,1\right].$

Let $\mu >0$ with $\mu \notin \mathbb{N}$ and $⌈\mu ⌉=n\in \mathbb{N}$. That is $0<1-n+\mu <1$, and let $f\in C^n\left(\left[-1,1\right]\right)$. Then, we have

$$D_{-1*}^{\mu }f\left(x\right):={}_{\left({\gamma }_j\right)\left({\omega }_j\right)}^{CA}{D}_{-1*}^{\mu ,\lambda }f\left(x\right):=$$

$$\frac{A\left(1-n+\mu \right)}{n-\mu }{\int }_{-1}^x{E}_{\left(1-n+\mu \right),\lambda }^{\left({\gamma }_j\right)}\left[\frac{-{\omega }_1\left(1-n+\mu \right)}{n-\mu }{\left(x-t\right)}^{1-n+\mu },\dots ,\right.$$

(35)

$$\left.\frac{-{\omega }_m\left(1-n+\mu \right)}{n-\mu }{\left(x-t\right)}^{1-n+\mu }\right]f^{\left(n\right)}\left(t\right)dt,$$

and

$$D_{1-}^{\mu }f\left(x\right):={}_{\left({\gamma }_j\right)\left({\omega }_j\right)}^{CA}{D}_{1-}^{\mu ,\lambda }f\left(x\right):=$$

$${\left(-1\right)}^n\frac{A\left(1-n+\mu \right)}{n-\mu }{\int }_x^1{E}_{\left(1-n+\mu \right),\lambda }^{\left({\gamma }_j\right)}\left[\frac{-{\omega }_1\left(1-n+\mu \right)}{n-\mu }{\left(t-x\right)}^{1-n+\mu },\dots ,\right.$$

(36)

$$\left.\frac{-{\omega }_m\left(1-n+\mu \right)}{n-\mu }{\left(t-x\right)}^{1-n+\mu }\right]f^{\left(n\right)}\left(t\right)dt,$$

∀$x\in \left[-1,1\right].$

We will set $D_{-1*}^{0}f=f$, $D_{1-}^{0}f=f$, and $D_{-1*}^{m}f=f^{\left(m\right)}$, $D_{1-}^{m}f={\left(-1\right)}^m{f}^{\left(m\right)}$, when $m\in \mathbb{N}.$

We make

Remark 1.

Fractional Calculi of Section 2.2 and Section 2.3 are special cases of abstract fractional calculus, see Section 2.1. In particular, the important condition (11) is fulfilled.

So, we will verify ${\int }_0^1{k}_h\left(z\right)dz\ge 1$, $h\ne 0.$

(I) First, for Section 2.2:

We notice that

$${\int }_0^1{z}^{N-\mu -1}{E}_{\rho ,N-\mu }^{-\gamma }\left(\omega z^{\rho }\right)dz=$$

(here $\rho ,N-\mu >0,$ $\gamma <0$, $\omega >0$)

$${\int }_0^1{z}^{N-\mu -1}\sum _{k=0}^{\infty }\frac{{\left(-\gamma \right)}_k}{k!\mathsf{\Gamma }\left(\rho k+N-\mu \right)}{\left(\omega z^{\rho }\right)}^{k}dz\stackrel{}{=}$$

$$\sum _{k=0}^{\infty }\frac{{\left(-\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+N-\mu \right)}{\int }_0^1{z}^{N-\mu -1}{z}^{\rho k}dz=$$

(37)

$$\sum _{k=0}^{\infty }\frac{{\left(-\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+N-\mu \right)}{\int }_0^1{z}^{\left(\rho k+N\right)-\mu -1}dz=$$

$$\sum _{k=0}^{\infty }\frac{{\left(-\gamma \right)}_k{\omega }^k}{k!\mathsf{\Gamma }\left(\rho k+N-\mu +1\right)}=E_{\rho ,N-\mu +1}^{-\gamma }\left(\omega \right)\ge 1,$$

for suitable $\omega >0.$

(II) Next, for Section 2.3:

Here ${\gamma }_j>0$, $j=1,\dots ,m;$ $\lambda =1;$ $\mathbb{N}/\ni \mu >0$, $⌈\mu ⌉=n\in \mathbb{N}$, ${\omega }_j<0$, $j=1,\dots ,m.$

Without loss of generality we assume that $A\left(1-n+\mu \right)>0.$

We have that

$$\frac{A\left(1-n+\mu \right)}{n-\mu }{\int }_0^1{E}_{\left(1-n+\mu \right),1}^{\left({\gamma }_j\right)}\left[\frac{-{\omega }_1\left(1-n+\mu \right)}{n-\mu }{z}^{1-n+\mu },\right.$$

$$\left.\dots ,\frac{-{\omega }_m\left(1-n+\mu \right)}{n-\mu }{z}^{1-n+\mu }\right]dz=$$

(here $0<1-\left(n-\mu \right)=1-n+\mu <1$)

$$\frac{A\left(1-n+\mu \right)}{n-\mu }{\int }_0^1\left[\sum _{k_1,\dots ,k_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{k_1}\dots {\left({\gamma }_m\right)}_{k_m}}{\mathsf{\Gamma }\left(1+\left({\displaystyle \sum _{j=1}^m}{k}_j\right)\left(1-n+\mu \right)\right)}\right.$$

$$\left.\frac{{\displaystyle \prod _{j=1}^m}\left({\left(\frac{-{\omega }_j\left(1-n+\mu \right)}{n-\mu }\right)}^{k_j}{z}^{\left(1-n+\mu \right){\displaystyle \sum _{j=1}^m}{k}_j}\right)}{k_1!\dots k_m!}\right]dz$$

$$=\left(\frac{A\left(1-n+\mu \right)}{n-\mu }\right)$$

$$\sum _{k_1,\dots ,k_m=0}^{\infty }\frac{{\left({\gamma }_1\right)}_{k_1}\dots {\left({\gamma }_m\right)}_{k_m}}{\mathsf{\Gamma }\left(1+\left({\displaystyle \sum _{j=1}^m}{k}_j\right)\left(1-n+\mu \right)\right)}\frac{{\displaystyle \prod _{j=1}^m}{\left(\frac{-{\omega }_j\left(1-n+\mu \right)}{n-\mu }\right)}^{k_j}}{k_1!\dots k_m!}{\int }_0^1{z}^{\left(1-n+\mu \right){\displaystyle \sum _{j=1}^m}{k}_j}dz$$

(38)

$$=\left(\frac{A\left(1-n+\mu \right)}{n-\mu }\right){\displaystyle \sum _{k_1,\dots ,k_m=0}^{\infty }}\frac{{\left({\gamma }_1\right)}_{k_1}\dots {\left({\gamma }_m\right)}_{k_m}}{\mathsf{\Gamma }\left(2+\left({\displaystyle \sum _{j=1}^m}{k}_j\right)\left(1-n+\mu \right)\right)}\frac{{\displaystyle \prod _{j=1}^m}{\left(\frac{-{\omega }_j\left(1-n+\mu \right)}{n-\mu }\right)}^{k_j}}{k_1!\dots k_m!}$$

$$=\frac{A\left(1-n+\mu \right)}{n-\mu }{E}_{\left(1-n+\mu \right),2}^{\left({\gamma }_j\right)}\left(\frac{-{\omega }_1\left(1-n+\mu \right)}{n-\mu },\dots ,\frac{-{\omega }_m\left(1-n+\mu \right)}{n-\mu }\right)\ge 1,$$

(39)

for suitable ${\omega }_j<0$, for $j=1,\dots ,m.$

We also need

Definition 3.

Let $f\in C^n\left(\left[-1,1\right]\right)$, $\mathbb{N}/\ni \mu >0$, $⌈\mu ⌉=n\in \mathbb{N};$ $\omega <0.$ That is $0<1-n+\mu <1$. The parametrized Caputo-Fabrizio non-singular kernel fractional derivatives, left and right of order μ, respectively, are given as follows (also see [16]):

$${}_{\omega }^{CF}{D}_{-1+}^{\mu }f\left(x\right):=\frac{1}{n-\mu }{\int }_{-1}^{x}exp\left(-\frac{\left(1-n+\mu \right)\omega }{n-\mu }\left(x-t\right)\right)f^{\left(n\right)}\left(t\right)dt,$$

(40)

$${}_{\omega }^{CF}{D}_{1-}^{\mu }f\left(x\right):=\frac{{\left(-1\right)}^n}{n-\mu }{\int }_x^{1}exp\left(-\frac{\left(1-n+\mu \right)\omega }{n-\mu }\left(t-x\right)\right)f^{\left(n\right)}\left(t\right)dt,$$

(41)

∀$x\in \left[-1,1\right].$

Equations (40) and (41) are special cases of (7) and (8).

We make

Remark 2.

We want to evaluate

$$\infty >{\int }_0^{1}exp\left(-\frac{\left(1-n+\mu \right)\omega }{n-\mu }z\right)dz$$

(call $\delta :=-\frac{\left(1-n+\mu \right)\omega }{n-\mu }$)

$$={\int }_0^1{e}^{\delta z}dz=\frac{1}{\delta }{e}^{\delta z}{|}_0^1=\frac{1}{\delta }\left(e^{\delta }-1\right)=\frac{e^{\delta }}{\delta }-\frac{1}{\delta }=\frac{{\displaystyle \sum _{k=0}^{\infty }}\frac{{\delta }^k}{k!}}{\delta }-\frac{1}{\delta }$$

(42)

$$=\sum _{k=0}^{\infty }\frac{{\delta }^{k-1}}{k!}=\sum _{k=0}^{\infty }\left(\frac{{\left(\frac{1-n+\mu }{n-\mu }\right)}^{k-1}{\left(-\omega \right)}^{k-1}}{k!}\right)\ge 1,$$

for suitable $\omega <0.$

So, again condition (11) is fulfilled.

3. Main Results

We give

Theorem 5.

Let $h,k,p$ be integers, $0\le h\le k\le p\in \mathbb{N}$ and let f be a real function, $f^{\left(p\right)}$ is continuous in $\left[-1,1\right]$ with modulus of continuity ${\omega }_1\left(f^{\left(p\right)},\delta \right)$, $\delta >0$. Let ${\alpha }_j\left(x\right)$, $j=h,h+1,\dots ,k$ be real functions, defined and bounded on $\left[-1,1\right]$ and assume for $x\in \left[0,1\right]$ that ${\alpha }_h\left(x\right)$ is either ≥ some number $\alpha >0$ or ≤ some number $\beta <0$. Let the real numbers ${\alpha }_0=0<{\alpha }_1<1<{\alpha }_2<2<\dots <{\alpha }_p<p$. Here, we adopt the abstract fractional calculus terminology and assumptions from above. So, ${}^{k_j}{D}_{*-1}^{{\alpha }_j}f$ stands for the abstract left Caputo type fractional derivative of order ${\alpha }_j$ anchored at $-1$. We consider the linear abstract left fractional differential operator

$$L:=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[{}^{k_j}{D}_{*-1}^{{\alpha }_j}\right]$$

(43)

and suppose, throughout $\left[0,1\right]$,

$$L\left(f\right)\ge 0.$$

Then, for any $n\in \mathbb{N}$, there exists a real polynomial $Q_n\left(x\right)$ of degree $\le n$ such that

$$L\left(Q_n\right)\ge 0\mathit{throughout}\left[0,1\right],$$

(44)

and

$$\underset{-1\le x\le 1}{max}\left|f\left(x\right)-Q_n\left(x\right)\right|\le C{n}^{k-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(45)

where C is independent of n or f.

Proof. 

Let $n\in \mathbb{N}$. By Theorem 2 given a real function g, with $g^{\left(p\right)}$ continuous in $\left[-1,1\right]$, there exists a real polynomial $q_n\left(x\right)$ of degree $\le n$ such that

$$\underset{-1\le x\le 1}{max}\left|g^{\left(j\right)}\left(x\right)-q_n^{\left(j\right)}\left(x\right)\right|\le R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right),$$

(46)

$j=0,1,\dots ,p,$ where $R_p$ is independent of n or g.

We notice that ($x\in \left[-1,1\right]$)

$$\left|\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}g\right)\left(x\right)-\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}{q}_n\right)\left(x\right)\right|=$$

$$\left|{\int }_{-1}^x{k}_j\left(x-t\right)g^{\left(j\right)}\left(t\right)dt-{\int }_{-1}^x{k}_j\left(x-t\right)q_n^{\left(j\right)}\left(t\right)dt\right|=$$

$$\left|{\int }_{-1}^x{k}_j\left(x-t\right)\left(g^{\left(j\right)}\left(t\right)-q_n^{\left(j\right)}\left(t\right)\right)dt\right|\le$$

$${\int }_{-1}^x{k}_j\left(x-t\right)\left|g^{\left(j\right)}\left(t\right)-q_n^{\left(j\right)}\left(t\right)\right|dt\stackrel{\left(46\right)}{\le }$$

(47)

$$\left({\int }_{-1}^x{k}_j\left(x-t\right)dt\right)R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right)=$$

$$\left({\int }_0^{x+1}{k}_j\left(z\right)dz\right)R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right)\le \left({\int }_0^2{k}_j\left(z\right)dz\right)R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right).$$

We have proved that

$$\left|\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}g\right)\left(x\right)-\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}{q}_n\right)\left(x\right)\right|\le$$

(48)

$$\left({\int }_0^2{k}_j\left(z\right)dz\right)R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right),\forall x\in \left[-1,1\right].$$

That is:

$$\underset{-1\le x\le 1}{max}\left|\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}g\right)\left(x\right)-\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}{q}_n\right)\left(x\right)\right|\le$$

$$\left({\int }_0^2{k}_j\left(z\right)dz\right)R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right),j=1,\dots ,p.$$

So, we have

$$\underset{-1\le x\le 1}{max}\left|\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}g\right)\left(x\right)-\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}{q}_n\right)\left(x\right)\right|\le {\lambda }_j{R}_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right),$$

(49)

where

$${\lambda }_j:={\int }_0^2{k}_j\left(z\right)dz,j=1,\dots ,p.$$

(50)

Inequality (49) is valid when $j=0$ by (46), and we can set ${\lambda }_0=1$.

Put

$$s_j\equiv \underset{-1\le x\le 1}{sup}\left|{\alpha }_h^{-1}\left(x\right){\alpha }_j\left(x\right)\right|,j=h,\dots ,k,$$

(51)

and

$${\eta }_n:=R_p{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)\left(\sum _{j=h}^k{s}_j{\lambda }_j{n}^{j-p}\right).$$

(52)

I. Suppose, throughout $\left[0,1\right]$, ${\alpha }_h\left(x\right)\ge \alpha >0$. Let $Q_n\left(x\right)$, $x\in \left[-1,1\right]$, be a real polynomial of degree $\le n$ so that

$$\underset{-1\le x\le 1}{max}\left|{}^{k_j}{D}_{*-1}^{{\alpha }_j}\left(f\left(x\right)+{\eta }_n{\left(h!\right)}^{-1}{x}^h\right)-\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}{Q}_n\right)\left(x\right)\right|\stackrel{\left(49\right)}{\le }$$

$${\lambda }_j{R}_p{n}^{j-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(53)

$j=0,1,\dots ,p.$

In particular, ($j=0$) holds

$$\underset{-1\le x\le 1}{max}\left|\left(f\left(x\right)+{\eta }_n{\left(h!\right)}^{-1}{x}^h\right)-Q_n\left(x\right)\right|\stackrel{\left(53\right)}{\le }{R}_p{n}^{-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(54)

and

$$\underset{-1\le x\le 1}{max}\left|f\left(x\right)-Q_n\left(x\right)\right|\le {\eta }_n{\left(h!\right)}^{-1}+R_p{n}^{-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)=$$

$${\left(h!\right)}^{-1}{R}_p{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)\left(\sum _{j=h}^k{s}_j{\lambda }_j{n}^{j-p}\right)+R_p{n}^{-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)\le$$

(55)

$$R_p{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)n^{k-p}\left(1+{\left(h!\right)}^{-1}\sum _{j=h}^k{s}_j{\lambda }_j\right).$$

That is:

$$\underset{-1\le x\le 1}{max}\left|f\left(x\right)-Q_n\left(x\right)\right|\le R_p\left(1+{\left(h!\right)}^{-1}\sum _{j=h}^k{s}_j{\lambda }_j\right)n^{k-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(56)

proving (45).

Here,

$$L=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[{}^{k_j}{D}_{*-1}^{{\alpha }_j}\right],$$

and suppose, throughout $\left[0,1\right]$, $Lf\ge 0$. So over $0\le x\le 1$, using (52) and (53), we have

$${\alpha }_h^{-1}\left(x\right)L\left(Q_n\left(x\right)\right)={\alpha }_h^{-1}\left(x\right)L\left(f\left(x\right)\right)+\frac{{\eta }_n}{h!}{}^{k_h}{D}_{*-1}^{{\alpha }_h}{x}^h+$$

$$\sum _{j=h}^k{\alpha }_h^{-1}\left(x\right){\alpha }_j\left(x\right)\left[{}^{k_j}{D}_{*-1}^{{\alpha }_j}{Q}_n\left(x\right)-{}^{k_j}{D}_{*-1}^{{\alpha }_j}f\left(x\right)-\frac{{\eta }_n}{h!}{}^{k_j}{D}_{*-1}^{{\alpha }_j}{x}^h\right]\ge$$

(57)

$$\frac{{\eta }_n}{h!}{}^{k_h}{D}_{*-1}^{{\alpha }_h}{x}^h-\left(\sum _{j=h}^k{s}_j{\lambda }_j{n}^{j-p}\right)R_p{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)=\frac{{\eta }_n}{h!}{}^{k_h}{D}_{*-1}^{{\alpha }_h}{x}^h-{\eta }_n=:\phi$$

(if $h=0$, then ${\alpha }_h=0$, and $\phi =0$).

If $h\ne 0$, then

$$\phi ={\eta }_n\left(\frac{{}^{k_h}{D}_{*-1}^{{\alpha }_h}{x}^h}{h!}-1\right)={\eta }_n\left({\int }_{-1}^x{k}_h\left(x-t\right)dt-1\right)=$$

$${\eta }_n\left({\int }_0^{x+1}{k}_h\left(z\right)dz-1\right)\ge {\eta }_n\left({\int }_0^1{k}_h\left(z\right)dz-1\right)\ge 0$$

(58)

by the assumption (11): ${\int }_0^1{k}_h\left(z\right)dz\ge 1$, when $h\ne 0.$

Hence, in both cases, we get

$$L\left(Q_n\left(x\right)\right)\ge 0,x\in \left[0,1\right].$$

(59)

II. Suppose, throughout $\left[0,1\right]$, ${\alpha }_h\left(x\right)\le \beta <0$. In this case let $Q_n\left(x\right)$, $x\in \left[-1,1\right]$, be a real polynomial of degree $\le n$ such that

$$\underset{-1\le x\le 1}{max}\left|{}^{k_j}{D}_{*-1}^{{\alpha }_j}\left(f\left(x\right)-{\eta }_n{\left(h!\right)}^{-1}{x}^h\right)-\left({}^{k_j}{D}_{*-1}^{{\alpha }_j}{Q}_n\right)\left(x\right)\right|\stackrel{\left(49\right)}{\le }$$

$${\lambda }_j{R}_p{n}^{j-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(60)

$j=0,1,\dots ,p.$

In particular, ($j=0$) holds

$$\underset{-1\le x\le 1}{max}\left|\left(f\left(x\right)-{\eta }_n{\left(h!\right)}^{-1}{x}^h\right)-Q_n\left(x\right)\right|\stackrel{\left(60\right)}{\le }{R}_p{n}^{-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(61)

and

$$\underset{-1\le x\le 1}{max}\left|f\left(x\right)-Q_n\left(x\right)\right|\le {\eta }_n{\left(h!\right)}^{-1}+R_p{n}^{-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)\stackrel{(\mathrm{as}\mathrm{before})}{\le }$$

(62)

$$R_p{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)n^{k-p}\left(1+{\left(h!\right)}^{-1}\sum _{j=h}^k{s}_j{\lambda }_j\right).$$

That is, (45) is again true.

Again suppose, throughout $\left[0,1\right]$, $Lf\ge 0$. Also if $0\le x\le 1$, then

$${\alpha }_h^{-1}\left(x\right)L\left(Q_n\left(x\right)\right)={\alpha }_h^{-1}\left(x\right)L\left(f\left(x\right)\right)-\frac{{\eta }_n}{h!}{}^{k_h}{D}_{*-1}^{{\alpha }_h}{x}^h+$$

$$\sum _{j=h}^k{\alpha }_h^{-1}\left(x\right){\alpha }_j\left(x\right)\left[{}^{k_j}{D}_{*-1}^{{\alpha }_j}{Q}_n\left(x\right)-{}^{k_j}{D}_{*-1}^{{\alpha }_j}f\left(x\right)+\frac{{\eta }_n}{h!}{}^{k_j}{D}_{*-1}^{{\alpha }_j}{x}^h\right]\stackrel{\left(60\right)}{\le }$$

$$-\frac{{\eta }_n}{h!}{}^{k_h}{D}_{*-1}^{{\alpha }_h}{x}^h+\left(\sum _{j=h}^k{s}_j{\lambda }_j{n}^{j-p}\right)R_p{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)=-\frac{{\eta }_n}{h!}{}^{k_h}{D}_{*-1}^{{\alpha }_h}{x}^h+{\eta }_n=:\psi$$

(if $h=0$, then ${\alpha }_h=0$, and $\psi =0$).

If $h\ne 0$, then

$$\psi ={\eta }_n\left[1-\frac{{}^{k_h}{D}_{*-1}^{{\alpha }_h}{x}^h}{h!}\right]={\eta }_n\left[1-{\int }_{-1}^x{k}_h\left(x-t\right)dt\right]=$$

(63)

$${\eta }_n\left[1-{\int }_0^{x+1}{k}_h\left(z\right)dz\right]\le {\eta }_n\left[1-{\int }_0^1{k}_h\left(z\right)dz\right]\le 0.$$

Hence, again, in both cases

$$L\left(Q_n\left(x\right)\right)\ge 0,\forall x\in \left[0,1\right].$$

(64)

□

We also present

Theorem 6.

Let $h,k,p$ be integers, $0\le h\le k\le p\in \mathbb{N},$ where h is even, and let f be a real function, $f^{\left(p\right)}$ is continuous in $\left[-1,1\right]$ with modulus of continuity ${\omega }_1\left(f^{\left(p\right)},\delta \right)$, $\delta >0$. Let ${\alpha }_j\left(x\right)$, $j=h,h+1,\dots ,k$ be real functions, defined and bounded on $\left[-1,1\right]$ and assume for $x\in \left[-1,0\right]$ that ${\alpha }_h\left(x\right)$ is either ≥ some number $\alpha >0$ or ≤ some number $\beta <0$. Let the real numbers ${\alpha }_0=0<{\alpha }_1<1<{\alpha }_2<2<\dots <{\alpha }_p<p$. Here, we adopt the abstract fractional calculus terminology and assumptions from above. So ${}^{k_j}{D}_{1-}^{{\alpha }_j}f$ stands for the abstract right Caputo type fractional derivative of order ${\alpha }_j$ anchored at 1. We consider the linear abstract right fractional differential operator

$$L:=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[{}^{k_j}{D}_{1-}^{{\alpha }_j}\right]$$

(65)

and suppose, throughout $\left[-1,0\right]$,

$$L\left(f\right)\ge 0.$$

Then, for any $n\in \mathbb{N}$, there exists a real polynomial $Q_n\left(x\right)$ of degree $\le n$ such that

$$L\left(Q_n\right)\ge 0\mathit{throughout}\left[-1,0\right],$$

(66)

and

$$\underset{-1\le x\le 1}{max}\left|f\left(x\right)-Q_n\left(x\right)\right|\le C{n}^{k-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(67)

where C is independent of n or f.

Proof. 

Let $x\in \left[-1,1\right]$, we observe that

$$\left|\left({}^{k_j}{D}_{1-}^{{\alpha }_j}g\right)\left(x\right)-\left({}^{k_j}{D}_{1-}^{{\alpha }_j}{q}_n\right)\left(x\right)\right|=$$

$$\left|{\int }_x^1{k}_j\left(t-x\right)g^{\left(j\right)}\left(t\right)dt-{\int }_x^1{k}_j\left(t-x\right)q_n^{\left(j\right)}\left(t\right)dt\right|=$$

$$\left|{\int }_x^1{k}_j\left(t-x\right)\left(g^{\left(j\right)}\left(t\right)-q_n^{\left(j\right)}\left(t\right)\right)dt\right|\le$$

$${\int }_x^1{k}_j\left(t-x\right)\left|g^{\left(j\right)}\left(t\right)-q_n^{\left(j\right)}\left(t\right)\right|dt\stackrel{\left(6\right)}{\le }$$

(68)

$$\left({\int }_x^1{k}_j\left(t-x\right)dt\right)R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right)=$$

$$\left({\int }_0^{1-x}{k}_j\left(z\right)dz\right)R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right)\le \left({\int }_0^2{k}_j\left(z\right)dz\right)R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right).$$

That is, we have derived

$$\underset{-1\le x\le 1}{max}\left|\left({}^{k_j}{D}_{1-}^{{\alpha }_j}g\right)\left(x\right)-\left({}^{k_j}{D}_{1-}^{{\alpha }_j}{q}_n\right)\left(x\right)\right|\le$$

$$\left({\int }_0^2{k}_j\left(z\right)dz\right)R_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right),j=1,\dots ,p.$$

(69)

We call

$${\lambda }_j:={\int }_0^2{k}_j\left(z\right)dz,j=1,\dots ,p.$$

(70)

Therefore, we can write

$$\underset{-1\le x\le 1}{max}\left|\left({}^{k_j}{D}_{1-}^{{\alpha }_j}g\right)\left(x\right)-\left({}^{k_j}{D}_{1-}^{{\alpha }_j}{q}_n\right)\left(x\right)\right|\le {\lambda }_j{R}_p{n}^{j-p}{\omega }_1\left(g^{\left(p\right)},\frac{1}{n}\right),$$

(71)

for $j=1,\dots ,p.$

Inequality (71) is valid when $j=0$ by (6), so we can set ${\lambda }_0=1$.

Put

$$s_j\equiv \underset{-1\le x\le 1}{sup}\left|{\alpha }_h^{-1}\left(x\right){\alpha }_j\left(x\right)\right|,j=h,\dots ,k,$$

(72)

and

$${\eta }_n:=R_p{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)\left(\sum _{j=h}^k{s}_j{\lambda }_j{n}^{j-p}\right).$$

(73)

I. Suppose, throughout $\left[-1,0\right]$, ${\alpha }_h\left(x\right)\ge \alpha >0$. Let $Q_n\left(x\right)$, $x\in \left[-1,1\right]$, be a real polynomial of degree $\le n$ so that

$$\underset{-1\le x\le 1}{max}\left|{}^{k_j}{D}_{1-}^{{\alpha }_j}\left(f\left(x\right)+{\eta }_n{\left(h!\right)}^{-1}{x}^h\right)-\left({}^{k_j}{D}_{1-}^{{\alpha }_j}{Q}_n\right)\left(x\right)\right|\stackrel{\left(71\right)}{\le }$$

$${\lambda }_j{R}_p{n}^{j-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(74)

$j=0,1,\dots ,p.$

In particular ($j=0$) holds

$$\underset{-1\le x\le 1}{max}\left|\left(f\left(x\right)+{\eta }_n{\left(h!\right)}^{-1}{x}^h\right)-Q_n\left(x\right)\right|\stackrel{\left(74\right)}{\le }{R}_p{n}^{-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(75)

and, as earlier,

$$\underset{-1\le x\le 1}{max}\left|f\left(x\right)-Q_n\left(x\right)\right|\le R_p\left(1+{\left(h!\right)}^{-1}\sum _{j=h}^k{s}_j{\lambda }_j\right)n^{k-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(76)

proving (67).

Here,

$$L=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[{}^{k_j}{D}_{1-}^{{\alpha }_j}\right],$$

and suppose, throughout $\left[-1,0\right]$, $Lf\ge 0$. So over $-1\le x\le 0$, we get

$${\alpha }_h^{-1}\left(x\right)L\left(Q_n\left(x\right)\right)={\alpha }_h^{-1}\left(x\right)L\left(f\left(x\right)\right)+\frac{{\eta }_n}{h!}{}^{k_h}{D}_{1-}^{{\alpha }_h}{x}^h+$$

$$\sum _{j=h}^k{\alpha }_h^{-1}\left(x\right){\alpha }_j\left(x\right)\left[{}^{k_j}{D}_{1-}^{{\alpha }_j}{Q}_n\left(x\right)-{}^{k_j}{D}_{1-}^{{\alpha }_j}f\left(x\right)-\frac{{\eta }_n}{h!}{}^{k_j}{D}_{1-}^{{\alpha }_j}{x}^h\right]\stackrel{\left(74\right)}{\ge }$$

(77)

$$\frac{{\eta }_n}{h!}{}^{k_h}{D}_{1-}^{{\alpha }_h}{x}^h-\left(\sum _{j=h}^k{s}_j{\lambda }_j{n}^{j-p}\right)R_p{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)=\frac{{\eta }_n}{h!}{}^{k_h}{D}_{1-}^{{\alpha }_h}{x}^h-{\eta }_n=:\xi$$

(if $h=0$, then ${\alpha }_h=0$, and $\xi =0$).

If $h\ne 0$, then

$$\xi ={\eta }_n\left(\frac{{}^{k_h}{D}_{1-}^{{\alpha }_h}{x}^h}{h!}-1\right)={\eta }_n\left({\left(-1\right)}^h{\int }_x^1{k}_h\left(t-x\right)dt-1\right)=$$

(h is even)

$${\eta }_n\left({\int }_x^1{k}_h\left(t-x\right)dt-1\right)={\eta }_n\left({\int }_0^{1-x}{k}_h\left(z\right)dz-1\right)\ge$$

$${\eta }_n\left({\int }_0^1{k}_h\left(z\right)dz-1\right)\ge 0,$$

(78)

by the assumption (11): ${\int }_0^1{k}_h\left(z\right)dz\ge 1$, when $h\ne 0.$

Hence, in both cases, we get

$$L\left(Q_n\left(x\right)\right)\ge 0,x\in \left[-1,0\right].$$

(79)

II. Suppose, throughout $\left[-1,0\right]$, ${\alpha }_h\left(x\right)\le \beta <0$. Let $Q_n\left(x\right)$, $x\in \left[-1,1\right]$, be a real polynomial of degree $\le n$ so that

$$\underset{-1\le x\le 1}{max}\left|{}^{k_j}{D}_{1-}^{{\alpha }_j}\left(f\left(x\right)-{\eta }_n{\left(h!\right)}^{-1}{x}^h\right)-\left({}^{k_j}{D}_{1-}^{{\alpha }_j}{Q}_n\right)\left(x\right)\right|\stackrel{(71)}{\le }$$

$${\lambda }_j{R}_p{n}^{j-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(80)

$j=0,1,\dots ,p.$

In particular ($j=0$) holds

$$\underset{-1\le x\le 1}{max}\left|\left(f\left(x\right)-{\eta }_n{\left(h!\right)}^{-1}{x}^h\right)-Q_n\left(x\right)\right|\stackrel{\left(80\right)}{\le }{R}_p{n}^{-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(81)

and, as earlier,

$$\underset{-1\le x\le 1}{max}\left|f\left(x\right)-Q_n\left(x\right)\right|\le R_p\left(1+{\left(h!\right)}^{-1}\sum _{j=h}^k{s}_j{\lambda }_j\right)n^{k-p}{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right),$$

(82)

proving (67).

Again, suppose, throughout $\left[-1,0\right]$, $Lf\ge 0$. Also if $-1\le x\le 0$, then

$${\alpha }_h^{-1}\left(x\right)L\left(Q_n\left(x\right)\right)={\alpha }_h^{-1}\left(x\right)L\left(f\left(x\right)\right)-\frac{{\eta }_n}{h!}{}^{k_h}{D}_{1-}^{{\alpha }_h}{x}^h+$$

$$\sum _{j=h}^k{\alpha }_h^{-1}\left(x\right){\alpha }_j\left(x\right)\left[{}^{k_j}{D}_{1-}^{{\alpha }_j}{Q}_n\left(x\right)-{}^{k_j}{D}_{1-}^{{\alpha }_j}f\left(x\right)+\frac{{\eta }_n}{h!}{}^{k_j}{D}_{1-}^{{\alpha }_j}{x}^h\right]\stackrel{\left(80\right)}{\le }$$

(83)

$$-\frac{{\eta }_n}{h!}{}^{k_h}{D}_{1-}^{{\alpha }_h}{x}^h+\left(\sum _{j=h}^k{s}_j{\lambda }_j{n}^{j-p}\right)R_p{\omega }_1\left(f^{\left(p\right)},\frac{1}{n}\right)=-\frac{{\eta }_n}{h!}{}^{k_h}{D}_{1-}^{{\alpha }_h}{x}^h+{\eta }_n$$

$$={\eta }_n\left(1-\frac{{}^{k_h}{D}_{1-}^{{\alpha }_h}{x}^h}{h!}\right)=:\rho$$

(if $h=0$, then ${\alpha }_h=0$, and $\rho =0$).

If $h\ne 0$, then

$$\rho ={\eta }_n\left(1-{\int }_x^1{k}_h\left(t-x\right)dt\right)={\eta }_n\left(1-{\int }_0^{1-x}{k}_h\left(z\right)dz\right)\le$$

(84)

$${\eta }_n\left(1-{\int }_0^1{k}_h\left(z\right)dz\right)\le 0.$$

Hence, in both cases, we get, again

$$L\left(Q_n\left(x\right)\right)\ge 0,\forall x\in \left[-1,0\right].$$

(85)

□

Conclusion 1.

Clearly Theorem 5 generalizes Theorem 1, and Theorem 6 generalizes Theorem 2.2, p. 12 of [4]. Furthermore, there, the approximating polynomial$Q_n$depends on$f,{\eta }_n,h;$which${\eta }_n$depends on n,$R_p,n,k,s_j,{\lambda }_j;$which${\lambda }_j$depends on$k_j$. I.e., polynomial$Q_n$among others depends on the type of fractional calculus we use.

Consequently, Theorem 5 is valid for the following left fractional linear differential operators:

(1)

$$L_1:=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[{}^C{D}_{\rho ,{\alpha }_j,\omega ,-1+}^{\gamma }\right],$$

(86)

where$\rho >0$,$\gamma <0$, and$\omega >0$large enough (from Prabhakar fractional calculus, see (16));

(2)

$$L_2:=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[D_{-1*}^{{\alpha }_j}\right],$$

(87)

(see (35)) where${\gamma }_j>0$,$j=1,\dots ,m;$$\lambda =1$; and small enough${\omega }_j<0$,$j=1,\dots ,m$(from generalized non-singular fractional calculus);

and (3)

$$L_3:=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[{}_{\omega }^{CF}{D}_{-1+}^{{\alpha }_j}\right],$$

(88)

with$\omega <0$, sufficiently small (from parametrized Caputo-Fabrizio non-singular kernel fractional calculus).

Similarly, Theorem 6 is valid for the following right fractional linear differential operators:

(1)${}^{*}$

$$L_1^{*}:=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[{}^C{D}_{\rho ,{\alpha }_j,\omega ,1-}^{\gamma }\right],$$

(89)

where$\rho >0$,$\gamma <0$, and$\omega >0$large enough (from Prabhakar fractional calculus, see (17));

(2)${}^{*}$

$$L_2^{*}:=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[D_{1-}^{{\alpha }_j}\right],$$

(90)

(see (36)) where${\gamma }_j>0$,$j=1,\dots ,m;$$\lambda =1$; and small enough${\omega }_j<0$,$j=1,\dots ,m$(from generalized non-singular fractional calculus);

and (3)${}^{*}$

$$L_3^{*}:=\sum _{j=h}^k{\alpha }_j\left(x\right)\left[{}_{\omega }^{CF}{D}_{1-}^{{\alpha }_j}\right],$$

(91)

with$\omega <0$, sufficiently small (from parametrized Caputo-Fabrizio non-singular kernel fractional calculus).

Our developed abstract fractional monotone approximation theory with its applications involves weaker conditions than the one with ordinary derivatives ([2]) and can cover many diverse general cases in a multitude of complex settings and environments.